Simulations of estimating causal effects using undersmoothed HAL
Seraphina Shi
2023-03-17
Simulation 2
Data structure: \(O = (W, A, Z, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Z - intermediate curve based on W and A
- Y - outcome, indicator of an event ?
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
- \(U_Z \sim Uniform(min = 0, max = 1)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0, max=5)\)
- \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
- \(Y = \mathbf{I}[U_Y < expit(-15 - 4W + 5A + Z(10 + 5W + 2AW))]\)
## W A Z Y
## Min. :-3.740827 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:-0.672122 1st Qu.:0.6269 1st Qu.:0.0000 1st Qu.:0.0000
## Median : 0.018596 Median :1.9994 Median :0.0000 Median :0.0000
## Mean : 0.005438 Mean :2.1095 Mean :0.4786 Mean :0.4324
## 3rd Qu.: 0.673552 3rd Qu.:3.3588 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. : 3.927491 Max. :5.0000 Max. :1.0000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.491 2.777 2.748 4.186 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.8917 2.1701 2.2565 3.5244 5.0000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.4486 1.8039 1.9338 3.0574 5.0000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.221 1.551 2.641 5.000
n = 200
CV HAL
results
## CV selected lambda (from one sample): 0.00342499231220454
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00387846621405548## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.000799982807508477
## which is 0.233572146909012 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.000851942858017283
## which is 0.217778357408167 * the average of 1000 lambda_CV## z=1:
## z=0:
CV vs Undersmoothing
## z=1:
## z=0:
Oevr a grid of lambda scalers
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.6451]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.6782]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.7113]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.7444]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.7775]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.8106]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.8437]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.8768]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.9099]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.9430]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.9761]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.10092]n = 500
CV HAL
results
## CV selected lambda (from one sample): 0.0016481890921567
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00184274700569968## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.000267629110962862
## which is 0.162377673918872 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.000393879557187377
## which is 0.213589118948609 * the average of 1000 lambda_CV## z=1:
## z=0:
CV vs Undersmoothing
## z=1:
## z=0:
Oevr a grid of lambda scalers
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.16449]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.16780]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.17111]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.17442]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.17773]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.18104]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.18435]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.18766]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.19097]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.19428]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.19759]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.20090]n = 1000
CV HAL
results
## CV selected lambda (from one sample): 0.00066416866564044
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00100946359885929## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.000320988082768554
## which is 0.483293023857175 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.000229565022553988
## which is 0.227979662379109 * the average of 1000 lambda_CV## z=1:
## z=0:
CV vs Undersmoothing
## z=1:
## z=0:
Oevr a grid of lambda scalers
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.26447]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.26778]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.27109]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.27440]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.27771]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.28102]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.28433]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.28764]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.29095]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.29426]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.29757]
## TableGrob (7 x 4) "arrange": 7 grobs
## z cells name grob
## 1 1 (2-3,2-3) arrange gtable[layout]
## 2 2 (4-5,1-2) arrange gtable[layout]
## 3 3 (4-5,3-4) arrange gtable[layout]
## 4 4 (6-7,1-2) arrange gtable[layout]
## 5 5 (6-7,3-4) arrange gtable[layout]
## 6 6 (3-3,4-4) arrange gtable[guide-box]
## 7 7 (1-1,1-4) arrange text[GRID.text.30088]